Question 1164909
Rules of inference and replacement can feel like a complex puzzle at first, but once you identify the "shape" of the statements, the rules become much clearer. Here are the step-by-step proofs for your three problems.

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### 1) Prove: 

**Given:** 

| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 |  | Premise |
| 2 |  | **Material Implication (Impl)** on 1 |
| 3 |  | **De Morgan's (DM)** on 2 |
| 4 |  | **Double Negation (DN)** on 3 |

**Explanation:** To break into a negated conditional, we first turn the "if-then" into an "or" statement, then use De Morgan's to distribute the negation.

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### 2) Prove: 

**Given:** 1. 
2. 

| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 |  | Premise |
| 2 |  | Premise |
| 3 |  | **Addition (Add)** on 2 |
| 4 |  | **Modus Ponens (MP)** on 1, 3 |
| 5 |  | **Simplification (Simp)** on 4 |

**Explanation:** Since we have , we can "add" anything to it using an "or" statement. This satisfies the left side of the conditional in Step 1, allowing us to extract the right side.

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### 3) Prove: 

**Given:**

1. 
2. 

| Step | Statement | Rule Used |
| --- | --- | --- |
| 1 |  | Premise |
| 2 |  | Premise |
| 3 |  | **De Morgan's (DM)** on 2 |
| 4 |  | **Disjunctive Syllogism (DS)** on 1, 3 |
| 5 |  | **Addition (Add)** on 4 |

**Explanation:** In Step 3, we used De Morgan's to show that neither  nor  is true. This contradicts the first part of Premise 1, which forces  to be true. Once we have , we can use Addition to add any variable (in this case, ).

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### Logic Tool Reference

A helpful tip for these is to always look at your **Conclusion** first. If it has a "" (like in problem 3), you often only need to find one side of it and use **Addition**.

Would you like to try another set, or should we go deeper into how **De Morgan's Law** works?